I use a Ruger 77 with a stainless Douglas barrel. Leupold 8x scope.
So how do you guys skin them? My grandmother was the best squirrel hunter I ever met, shoe showed me how she did it. Grab the tail and start cutting through the hide right at the tail/back junction. Feather the hide down the back a bit, then step on the tail, and pull up with the back legs. It will skin right to the front legs. I carry a knife, plastic bag and side cutters in a bag, and skin and clean in the woods when I get a few. Much easier when they are warm!
I have skinned them that way too, sometimes it comes off just like you explained, sometimes not so much. I've been thinking about the air compressor trick to make it even easier.
